A First Look Into It
In math we have and use numbers. Numbers represent the value a thing or a bunch of things may have. Sometimes we may not know what numbers are present somewhere. For instance, we may have to fit in boxes into a... a car. We know we can't put in 400 boxes because we can even fit that in a small office. But we can't decide if it can take 20, or if there is a way to put that in. So we have 12 by 12 inch boxes, and we have 20 of them is all we know. But we don't have the number of boxes we require to be ready with when the car comes around. To find out that we need the size of the space we putting them in.
The numbers we are using here are lengths and counts, and they are all 'variables'. A variable is a number which we can choose its value or which value can change. The size of the boxes are 'known variables' and the counts of the boxes are 'unknown variables'. We use known variables to find unknown ones. That is why algebra is used. There are also constants which are numbers that won't change through out the calculation. One such constant is, may the amount of things to be put in the boxes, because Brock the bully, won't just forget useless stuff.
In math, letters x, y, z are usually used to represent variables. You must have enough knowns to find any number of unknowns, or you must have enough equations or relationships between knowns and unknowns to find the unknowns.
The One Unknowns Dimensions
If you need to find one unknown, then you must have one equation at least, that has one unknown at most, and any number or constants and/or coefficients. The way the relation looks is also important. For instance, look at these equations:
y=3;
y+x2=4
exp(x)+y2=4
You can see that the the second equation is going to give you more problems. The first one you can solve simply by looking at it. Or you could do the math:
y=3;
y+x2=4
x2 =4-y=4-3
x2=1
x=1
Incase you were wondering, x2 means 'x raised to the power of 2' or 'x squared'.
An equation with one unknown and one variable is also possible such as:
x2+3x-1=0
cos(x)+sin(x)=0
In each case there is one unknown but the relation has a problem that the variable is processed in a way as to bring out an answer but we can't find out how. The equations we will deal with here are the linear and the quadratic equations and we will use only x. Linear equations are equations in which the highest power is 1 while quadratic equations are equations where the highest power is 2.
For more math lessons on graphs and sets: |General Graphing Sets
Linear Equations
How do we solve linear equations? We need an example:
5x - 4 = 8x + 2
We can treat the components '5x' and '8x' as a kind of number, and '4' and '2' as another kind of number. The numbers with 'x' on them we can add or subtract from each other, but we can't add or subtract them with the numbers without any 'x' on them.
5x - 4 = 8x + 2
5x-8x = 2+4
3x=6
When you move a number about the equals sign, you must change its sign from positive to negative or vice versa. So '+8x' becomes '-8x' and '-4' becomes '+4'. You see, you carry numbers along with the sign in front of them, so that the '2' is actually '+2'. If no sign is before a number like in '5', it a positive number, '+5'. So, what will you multiply with 3 to give 6? To know that, you divclasse both sclasses by '3', since '3' is '3 times something equals 6'.
3x/3=6/3
x=2;
So the unknown's value is 2 or x is equal to 2.
Quadratic Equations
Quadratic equations have a highest power of 2. One such equation is:
5x2-16x+3=0
How you solve this is to look for 'common factors'. A 'common factor' is a number or variable that multiplies a set of additions and subtractions, like in:
4x-8y+12
-3xy + 5x2y3 - xy2
Remember, when a number comes after a letter, it stands for a power. So 'y2' is ' y-squared'. When two a number is next to a letter or two letter are next to each other it is a multiplication. So 4x means '4 times x' and xy means 'x times y' and x2y3 means 'x-squared times y raised to the third power'.
The common factor in the first equation is 4. The common factor in the second equation is xy. So e have for those two equations:
4(x-2y+3)
xy(-3+5xy2 -y)
How you look for common factors in the quadratic equation above is to cut '-16x' such that you can find a suitable factor for '5x2' and for 3.You first multiply 5 with 3 which gives 15, and you say ' what two numbers when I times them gives me 15 and when I add them gives me -16?' and you see '-15 and -1'. It is not always simple to do. You can see that
-15 + (-1)=-15-1=-16
-15 times -1 =15
So we rewrite the equation:
5x2 -15x-x +3=0
5x(x-3)-1(x-3)=0
It now seems like we have a common factor of '(x-3)'. When we take that out, we have:
(x-3)(5x-1)=0
If two unknown numbers are multiplied together, and the answer is zero, one or both numbers must be equal to zero. That's about all you got. So either
(x-3)=0
or (5x-1)=0
so x=3 or x=1/5
The number of answers is equal to the highest power. For linear equations, there is one answer and for quadratic equations there are two answers. There is a formula for solving Quadratic equations. You can search for it in you textbooks or online. All you have to do is plug in the numbers into this equation and pull out the answers.
The Two Unknowns Dimensions
When dealing with two unknowns, we need at least two equations. Again we will deal with simply x and y and not special functions like sine or cosine. Say we had the following two equations:
4y+5x=1
-y+4x=0
The way to solve these equations is to try to remove one unknown so you could solve the other one. There are two simple ways:
- Substitution
- Eliminating
Substitution
To use substitution method you take one equation and place one variable on the left side of the equals sign and place the rest of the factors on the other side, This variable will be substituted in the other equation like this:
using the second equation
-y+4x=0
-y=-4x
y=4x
substitute for y in the first equation
4(4x)+5x=1
16x+5x=1
21x=1
x=1/21
Eliminating
With this method you find multiplying factors for both equations, that produce terms that are exactly alike in both equations. You the eliminate these leaving you with a term containg one unknown only
4y+5x=1
-y+4x=0
multiply the first one with 4 and the second one with -1
making the y-terms equal
-4y-5x=-1
-4y+16x=0
You now 'add' or 'subtract' the first one from the second, depend on which will eliminate the '-4y'. Since they both have equal signs we subtract.
-4y-5x=-1
-4y+16x=0
_______________
-21x = -1
x=1/21
The Three Unknowns Dimensions
When three unknowns are involved, things get complex.
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